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Theorem plyconst 26127

Description: A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)

**Assertion**
Ref	Expression
plyconst	⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))

Proof of Theorem plyconst Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exp0 13990	. . . . . . 7 ⊢ (𝑧 ∈ ℂ → (𝑧↑0) = 1)
2	1	adantl 481	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ ℂ) → (𝑧↑0) = 1)
3	2	oveq2d 7369	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) = (𝐴 · 1))
4		ssel2 3932	. . . . . . 7 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ ℂ)
5	4	adantr 480	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ)
6	5	mulridd 11151	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ ℂ) → (𝐴 · 1) = 𝐴)
7	3, 6	eqtrd 2764	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) = 𝐴)
8	7	mpteq2dva 5188	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0))) = (𝑧 ∈ ℂ ↦ 𝐴))
9		fconstmpt 5685	. . 3 ⊢ (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ 𝐴)
10	8, 9	eqtr4di 2782	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0))) = (ℂ × {𝐴}))
11		0nn0 12417	. . 3 ⊢ 0 ∈ ℕ₀
12		eqid 2729	. . . 4 ⊢ (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0))) = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0)))
13	12	ply1term 26125	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆 ∧ 0 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0))) ∈ (Poly‘𝑆))
14	11, 13	mp3an3 1452	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑0))) ∈ (Poly‘𝑆))
15	10, 14	eqeltrrd 2829	1 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3905 {csn 4579 ↦ cmpt 5176 × cxp 5621 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 0cc0 11028 1c1 11029 · cmul 11033 ℕ₀cn0 12402 ↑cexp 13986 Polycply 26105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fz 13429 df-fzo 13576 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-ply 26109

This theorem is referenced by: ply0 26129 plysub 26140 plyco 26162 0dgr 26166 coemulc 26176 coesub 26178 dgrmulc 26193 dgrsub 26194 plyremlem 26228 fta1lem 26231 vieta1lem2 26235 qaa 26247 iaa 26249 taylply2 26291 taylply2OLD 26292 dchrfi 27182 mpaaeu 43123 rngunsnply 43142 cjnpoly 46874

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